117 ideas
| 12865 | Analytic philosophers may prefer formal systems because natural language is such mess [Simons] |
| Full Idea: The untidiness of natural language in its use of 'part' is perhaps one of the chief reasons why mereolologists have preferred to investigate formal systems with nice algebraic properties rather than get out and mix it with reality in all its messiness. | |
| From: Peter Simons (Parts [1987], 6.4) | |
| A reaction: [See Idea 12864 for the uses of 'part'] I am in the unhappy (and probably doomed) position of wanting to avoid both approaches. I try to operate as if the English language were transparent and we can just discuss the world. Very naïve. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 12815 | Classical mereology doesn't apply well to the objects around us [Simons] |
| Full Idea: The most fundamental criticism of classical mereology is that the theory is not applicable to most of the objects around us, and is accordingly of little use as a formal reconstruction of the concepts of part and whole which we actually employ. | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: This sounds splendidly dismissive, but one might compare it with possible worlds semantics for modal logic, which most people take with a pinch of salt as an actual commitment, but find wonderfully clarifying in modal reasoning. |
| 12819 | A 'part' has different meanings for individuals, classes, and masses [Simons] |
| Full Idea: It emerges that 'part', like other formal concepts, is not univocal, but has analogous meanings according to whether we talk of individuals, classes, or masses. | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: He suggests that unrestricted sums are appropriate for the last two, but not for individuals. There must be something univocal about the word - some awareness of a possible whole or larger entity to which the thing could belong. |
| 12832 | Complement: the rest of the Universe apart from some individual, written x-bar [Simons] |
| Full Idea: The 'complement' of each individual in mereology is the rest of the Universe outside it, that is U - x, but written as x-bar [x with a horizontal bar above it]. | |
| From: Peter Simons (Parts [1987], 1.1.10) | |
| A reaction: [Don't have a font for x-bar] See Idea 12831 for the 'Universe'. Simons suggest that the interest of this term is mainly historical and algebraic. |
| 12834 | Criticisms of mereology: parts? transitivity? sums? identity? four-dimensional? [Simons] |
| Full Idea: Main criticisms of mereology: we don't mean 'part' as improper; transitivity of 'part' is sometimes not transitive; no guarantee that there are 'sums'; the identity criteria for individuals are false; we are forced into materialistic four-dimensionalism. | |
| From: Peter Simons (Parts [1987], 3.2) | |
| A reaction: [Compressed summary; for four-dimensionalism see under 'Identity over Time'] Simons says these are in ascending order of importance. |
| 12827 | Difference: the difference of individuals is the remainder of an overlap, written 'x - y' [Simons] |
| Full Idea: The 'difference' of two individuals is the largest individual contained in x which has no part in common with y, expressed by 'x - y', read as 'the difference of x and y'. | |
| From: Peter Simons (Parts [1987], 1.1.07) |
| 12828 | General sum: the sum of objects satisfying some predicate, written σx(Fx) [Simons] |
| Full Idea: The 'general sum' of all objects satisfying a certain predicate is denoted by a variable-binding operator, expressed by 'σx(Fx)', read as 'the sum of objects satisfying F'. | |
| From: Peter Simons (Parts [1987], 1.1.08) | |
| A reaction: This, it seems, is introduced to restrict some infinite classes which aspire to be sums. |
| 12822 | Proper or improper part: x < y, 'x is (a) part of y' [Simons] |
| Full Idea: A 'proper or improper part' is expressed by 'x < y', read as 'x is (a) part of y'. The relatively minor deviation from normal usage (of including an improper part, i.e. the whole thing) is warranted by its algebraical convenience. | |
| From: Peter Simons (Parts [1987], 1.1.02) | |
| A reaction: Including an improper part (i.e. the whole thing) is not, Simons points out, uncontroversial, because the part being 'equal' to the whole is read as being 'identical' to the whole, which Simons is unwilling to accept. |
| 12823 | Overlap: two parts overlap iff they have a part in common, expressed as 'x o y' [Simons] |
| Full Idea: Two parts 'overlap' mereologically if and only if they have a part in common, expressed by 'x o y', read as 'x overlaps y'. Overlapping is reflexive and symmetric but not transitive. | |
| From: Peter Simons (Parts [1987], 1.1.03) | |
| A reaction: Simons points out that we are uncomfortable with overlapping (as in overlapping national boundaries), because we seem to like conceptual boundaries. We avoid overlap even in ordering primary colour terms, by having a no-man's-land. |
| 12824 | Disjoint: two individuals are disjoint iff they do not overlap, written 'x | y' [Simons] |
| Full Idea: Two individuals are 'disjoint' mereologically if and only if they do not overlap, expressed by 'x | y', read as 'x is disjoint from y'. Disjointedness is symmetric. | |
| From: Peter Simons (Parts [1987], 1.1.04) |
| 12825 | Product: the product of two individuals is the sum of all of their overlaps, written 'x · y' [Simons] |
| Full Idea: For two overlapping individuals their 'product' is the individual which is part of both and such that any common part of both is part of it, expressed by 'x · y', read as 'the product of x and y'. | |
| From: Peter Simons (Parts [1987], 1.1.05) | |
| A reaction: That is, the 'product' is the sum of any common parts between two individuals. In set theory all sets intersect at the null set, but mereology usually avoids the 'null individual'. |
| 12826 | Sum: the sum of individuals is what is overlapped if either of them are, written 'x + y' [Simons] |
| Full Idea: The 'sum' of two individuals is that individual which something overlaps iff it overlaps at least one of x and y, expressed by 'x + y', read as 'the sum of x and y'. It is central to classical extensional mereologies that any two individuals have a sum. | |
| From: Peter Simons (Parts [1987], 1.1.06) | |
| A reaction: This rather technical definition (defining an individual by the possibility of it being overlapped) does not always coincide with the smallest individual containing them both. |
| 12829 | General product: the nucleus of all objects satisfying a predicate, written πx(Fx) [Simons] |
| Full Idea: The 'general product' or 'nucleus' of all objects satisfying a certain predicate is denoted by a variable-binding operator, expressed by 'πx(Fx)', read as 'the product of objects satisfying F'. | |
| From: Peter Simons (Parts [1987], 1.1.08) | |
| A reaction: See Idea 12825 for 'product'. 'Nucleus' is a helpful word here. Thought: is the general product a candidate for a formal definition of essence? It would be a sortal essence - roughly, what all beetles have in common, just by being beetles. |
| 12830 | Universe: the mereological sum of all objects whatever, written 'U' [Simons] |
| Full Idea: The 'Universe' in mereology is the sum of all objects whatever, a unique individual of which all individuals are part. This is denoted by 'U'. Strictly, there can be no 'empty Universe', since the Universe is not a container, but the whole filling. | |
| From: Peter Simons (Parts [1987], 1.1.09) | |
| A reaction: This, of course, contrasts with set theory, which cannot have a set of all sets. At the lower end, set theory does have a null set, while mereology has no null individual. See David Lewis on combining the two theories. |
| 12831 | Atom: an individual with no proper parts, written 'At x' [Simons] |
| Full Idea: An 'atom' in mereology is an individual with no proper parts. We shall use the expression 'At x' to mean 'x is an atom'. | |
| From: Peter Simons (Parts [1987], 1.1.11) | |
| A reaction: Note that 'part' in standard mereology includes improper parts, so every object has at least one part, namely itself. |
| 12844 | Dissective: stuff is dissective if parts of the stuff are always the stuff [Simons] |
| Full Idea: Water is said not to be 'dissective', since there are parts of any quantity of water which are not water. | |
| From: Peter Simons (Parts [1987], 4.2) | |
| A reaction: This won't seem to do for any physical matter, but presumably parts of numbers are always numbers. |
| 12816 | Classical mereology doesn't handle temporal or modal notions very well [Simons] |
| Full Idea: The underlying logic of classical extensional mereology does not have the resources to deal with temporal and modal notions such as temporary part, temporal part, essential part, or essential permanent part. | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: Simons tries to rectify this in the later chapters of his book, with modifications rather than extensions. Since everyone struggles with temporal and modal issues of identity, we shouldn't judge too harshly. |
| 12821 | The part-relation is transitive and asymmetric (and thus irreflexive) [Simons] |
| Full Idea: Formally, the part-relation is transitive and asymmetric (and thus irreflexive). Hence nothing is a proper part of itself, things aren't proper parts of one another, and if one is part of two which is part of three then one is part of three. | |
| From: Peter Simons (Parts [1987], 1.1.1) |
| 18847 | Each wheel is part of a car, but the four wheels are not a further part [Simons] |
| Full Idea: The four wheels of a car are parts of it (each is part of it), but there is not a fifth part consisting of the four wheels. | |
| From: Peter Simons (Parts [1987], 4.6) | |
| A reaction: This raises questions about the transitivity of parthood. If there are parts of parts of wholes, the basic parts are OK, and the whole is OK, but how can there also be an intermediate part? Try counting the parts of this whole! |
| 12813 | Two standard formalisations of part-whole theory are the Calculus of Individuals, and Mereology [Simons] |
| Full Idea: The standardly accepted formal theory of part-whole is classical extensional mereology, which is known in two logical guises, the Calculus of Individuals of Leonard and Goodman, and the Mereology of Lesniewski. | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: Simons catalogues several other modern attempts at axiomatisation in his chapter 2. |
| 12848 | The same members may form two groups [Simons] |
| Full Idea: Groups may coincide in membership without being identical - extensionality goes. | |
| From: Peter Simons (Parts [1987], 4.9) | |
| A reaction: Thus an eleven-person orchestra may also constitute a football team. What if a pile of stones is an impediment to you, and useful to me? Is it then two groups? Suppose they hum while playing football? (Don't you just love philosophy?) |
| 12846 | A 'group' is a collection with a condition which constitutes their being united [Simons] |
| Full Idea: We call a 'collection' of jewels a 'group' term. Several random musicians are unlikely to be an orchestra. If they come together regularly in a room to play, such conditions are constitutive of an orchestra. | |
| From: Peter Simons (Parts [1987], 4.4) | |
| A reaction: Clearly this invites lots of borderline cases. Eleven footballers don't immediately make a team, as followers of the game know well. |
| 12861 | 'The wolves' are the matter of 'the pack'; the latter is a group, with different identity conditions [Simons] |
| Full Idea: 'The wolves' is a plural term referring to just these animals, whereas 'the pack' of wolves refers to a group, and the group and plurality, while they may coincide in membership, have different identity conditions. The wolves are the matter of the pack. | |
| From: Peter Simons (Parts [1987], 6.4) | |
| A reaction: Even a cautious philosopher like Simons is ready to make bold ontological commitment to 'packs', on the basis of something called 'identity conditions'. I think it is just verbal. You can qualify 'the wolves' and 'the pack' to make them identical. |
| 12876 | Philosophy is stuck on the Fregean view that an individual is anything with a proper name [Simons] |
| Full Idea: Modern philosophy is still under the spell of Frege's view that an individual is anything that has a proper name. (Note: But not only are empty names now recognised, but some are aware of the existence of plural reference). | |
| From: Peter Simons (Parts [1987], 8.1) | |
| A reaction: Presumably every electron in the universe is an individual, and every (finite) number which has never been named has a pretty clear identity. Presumably Pegasus, John Doe, and 'the person in the kitchen' have to be accommodated. |
| 10429 | It is best to say that a name designates iff there is something for it to designate [Sainsbury] |
| Full Idea: It is better to say that 'For all x ("Hesperus" stands for x iff x = Hesperus)', than to say '"Hesperus" stands for Hesperus', since then the expression can be a name with no bearer (e.g. "Vulcan"). | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.2) | |
| A reaction: In cases where it is unclear whether the name actually designates something, it seems desirable that the name is at least allowed to function semantically. |
| 10425 | Definite descriptions may not be referring expressions, since they can fail to refer [Sainsbury] |
| Full Idea: Almost everyone agrees that intelligible definite descriptions may lack a referent; this has historically been a reason for not counting them among referring expressions. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.2) | |
| A reaction: One might compare indexicals such as 'I', which may be incapable of failing to refer when spoken. However 'look at that!' frequently fails to communicate reference. |
| 10438 | Definite descriptions are usually rigid in subject, but not in predicate, position [Sainsbury] |
| Full Idea: Definite descriptions used with referential intentions (usually in subject position) are normally rigid, ..but in predicate position they are normally not rigid, because there is no referential intention. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.5) | |
| A reaction: 'The man in the blue suit is the President' seems to fit, but 'The President is the head of state' doesn't. Seems roughly right, but language is always too complex for philosophers. |
| 12845 | Some natural languages don't distinguish between singular and plural [Simons] |
| Full Idea: The syntactic distinction between singular and plural is not a universal feature of natural languages. Chinese manages nicely without it, and Sanskrit makes a tripartite distinction between singular, dual, and plural (more than two). | |
| From: Peter Simons (Parts [1987], 4.3) | |
| A reaction: Simons is mounting an attack on the way in which modern philosophy and logic has been mesmerised by singular terms and individuated objects. Most people seem now to agree with Simons. There is stuff, as well as plurals. |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 12838 | Four-dimensional ontology has no change, since that needs an object, and time to pass [Simons] |
| Full Idea: In the four-dimensional ontology there may be timeless variation, but there is no change. Change consists in an object having first one property and then another contrary one. But processes all have their properties timelessly. | |
| From: Peter Simons (Parts [1987], 3.4) | |
| A reaction: Possibly Simons is begging the question here. The phenomena which are traditionally labelled as 'change' are all nicely covered in the four-D account. Change is, we might say, subsumed in the shape of the space-time 'worm'. |
| 12842 | There are real relational changes, as well as bogus 'Cambridge changes' [Simons] |
| Full Idea: It is a mistake to call bogus Cambridge changes 'relational changes', since there are real relational changes, such as the changes in the relative positions and distances of several bodies. | |
| From: Peter Simons (Parts [1987], 4.1) | |
| A reaction: I'm not sure how you distinguish the two. If we swap seats, that is a real change. If everyone moves away from where I am sitting, is that real or Cambridge? If I notice, I might be upset, but suppose I don't notice? Nothing about me changes. |
| 12841 | I don't believe in processes [Simons] |
| Full Idea: I have been unable to see that there are processes. | |
| From: Peter Simons (Parts [1987], 4.1 n4) | |
| A reaction: My problem here is that I am inclined to think of the mind as a process of the brain. The fact that a reductive account can be given of a process doesn't mean that we can deny there existence. Is there no such thing as decay, or erosion? |
| 12836 | Fans of process ontology cheat, since river-stages refer to 'rivers' [Simons] |
| Full Idea: Proponents of process ontology (except perhaps Whitehead, who is obscure) indulge in double-talk with concrete examples. It is cheating to talk of 'cat-processes', or 'bathing in river-stages'. You can't change the subject and leave the predicate alone. | |
| From: Peter Simons (Parts [1987], 3.4) | |
| A reaction: It is one thing to admit processes into one's ontology, and another to have a 'process ontology', which presumably reduces objects to processes. I suppose the interest of continuant objects is precisely the aspect of them that is above any process. |
| 12880 | Moments are things like smiles or skids, which are founded on other things [Simons] |
| Full Idea: A 'moment' is something which is founded on something else. Examples are legion: smiles, headaches, gestures, skids, collisions, fights, thought, all founded on their participants, the continuants involved in them. | |
| From: Peter Simons (Parts [1987], 8.4) | |
| A reaction: The idea of a 'moment' and 'foundation' come from Husserl Log. Inv. 3. Simons says moments 'have a bright future in ontology'. It would be better if fewer of his examples involved human beings and their perceptions. |
| 12881 | A smiling is an event with causes, but the smile is a continuant without causes [Simons] |
| Full Idea: A smiling, being an event, has causes and effects, whereas the smile thereby produced is a continuant, and has itself neither causes nor effects. | |
| From: Peter Simons (Parts [1987], 8.5) | |
| A reaction: This is dogmatic, hopeful and a bit dubious. Simons is very scathing about processes in ontology. There seem to be two descriptions, with distinctive syntax, but it is hard to believe that in reality we have two types of thing present. |
| 12883 | Moving disturbances are are moments which continuously change their basis [Simons] |
| Full Idea: Moving disturbances are a special and interesting kind of continuant: moments which continuously change their fundaments. | |
| From: Peter Simons (Parts [1987], 8.5) | |
| A reaction: [a smile is a moment, and the face its fundament] I'm thinking he's got this wrong. Compare Idea 12882. Disturbances can't be continuants, because the passing of time is essential to them, but not to a continuant. |
| 12882 | A wave is maintained by a process, but it isn't a process [Simons] |
| Full Idea: A wave is maintained by a process transferring motion from particle to particle of the medium, but it is not identical with this process. | |
| From: Peter Simons (Parts [1987], 8.5) | |
| A reaction: I'm inclined to think of the mind as a process. There are some 'things' which only seem to exist if they have a duration. Bricks can be instantaneous, but minds and waves can't. A wave isn't a continuant. A hill isn't a wave. |
| 12840 | I do not think there is a general identity condition for events [Simons] |
| Full Idea: Like Anscombe (1979) I do not think there is such a creature as a general identity condition for events. | |
| From: Peter Simons (Parts [1987], 4.1 n1) | |
| A reaction: My working definition of an event is 'any part of a process which can be individuated'. This leaves you trying to define a process, and define individuate, and then to realise that individuation is not an objective matter. |
| 12839 | Relativity has an ontology of things and events, not on space-time diagrams [Simons] |
| Full Idea: A closer examination of the concepts and principles of relativity shows that they rest squarely on an ontology of things and events (not on convenient 'space-time diagrams'). Acceleration concerns non-zero mass, but only continuants can have a mass. | |
| From: Peter Simons (Parts [1987], 3.4) | |
| A reaction: The point here is that fans of four-dimensionalism like to claim that they are more in touch with modern physics, because 'time is just another dimension, like space, so objects are spread across it'. Simons sounds right about this. |
| 12879 | Independent objects can exist apart, and maybe even entirely alone [Simons] |
| Full Idea: An object a is ontologically independent of b if a can exist without b, if there is a possible world in which in which a exists and b does not. In the strongest sense, an object is independent if it could be all there is. | |
| From: Peter Simons (Parts [1987], 8.4) | |
| A reaction: Simons calls the strongest version a 'startling' one which maybe not even God could achieve. |
| 12847 | Mass nouns admit 'much' and 'a little', and resist 'many' and 'few'. [Simons] |
| Full Idea: Syntactic criteria for mass nouns include that they admit 'much' and 'a little', and resist 'many' and 'few'. | |
| From: Peter Simons (Parts [1987], 4.6) | |
| A reaction: That is, they don't seem to be countable. Sortal terms are those which pick out countables. |
| 12863 | Mass terms (unlike plurals) are used with indifference to whether they can exist in units [Simons] |
| Full Idea: Mass terms and plural terms differ principally in the indifference of mass terms to matters of division. A mass term can be used irrespective of how, indeed whether, the denotatum comes parcelled in units. | |
| From: Peter Simons (Parts [1987], 6.4) | |
| A reaction: It seems more to the point to say that mass terms (stuff) don't need units to exist, and you can disperse the units (the cups of water) without affecting the identity of the stuff. You can't pulverise a pile of stones and retain the stones. |
| 12862 | Gold is not its atoms, because the atoms must be all gold, but gold contains neutrons [Simons] |
| Full Idea: The mass of gold cannot be identified with the gold atoms, because whatever is part of the gold atoms is gold, whereas not every part of the gold is gold (for example, the neutrons in it are not gold). | |
| From: Peter Simons (Parts [1987], 6.4) | |
| A reaction: There is something too quick about arguments like this. It comes back to nominal v real essence. We apply 'gold' to the superficial features of the stuff, but deep down we may actually mean the atomic structure. See Idea 12812. |
| 12859 | A mixture can have different qualities from its ingredients. [Simons] |
| Full Idea: The qualities of a mixture need not be those of its ingredients in isolation. | |
| From: Peter Simons (Parts [1987], 6.2) | |
| A reaction: It depends on what you mean by a quality. Presumably we can give a reductive account of the qualities of the mixture, as long as no reaction has taken place. The taste of a salad is just the sum of its parts. |
| 12858 | Mixtures disappear if nearly all of the mixture is one ingredient [Simons] |
| Full Idea: If a cupful of dirty water is mixed evenly with a ton of earth, no dirty water remains, and the same goes if we mix it evenly with a lake of clean water. | |
| From: Peter Simons (Parts [1987], 6.2) | |
| A reaction: This means that a mixture is a vague entity, subject to the sorites paradox. If the dirt was cyanide, we would consider the water to be polluted by it down to a much lower level. |
| 12850 | To individuate something we must pick it out, but also know its limits of variation [Simons] |
| Full Idea: We have not finished deciding what Fido is when we can pick him out from his surroundings at any one time. ...Knowing what Fido is depends on knowing roughly within what limits his flux of parts is tolerable. | |
| From: Peter Simons (Parts [1987], 5.2) | |
| A reaction: I like this. We don't know the world until we know its modal characteristics (its powers or dispositions). Have you 'individuated' a hand grenade if you think it is a nice ornament? |
| 12860 | Sortal nouns for continuants tell you their continuance- and cessation-conditions [Simons] |
| Full Idea: A sortal noun for a kind of continuant tells us, among other things, under what conditions the object continues to exist and under what conditions it ceases to exist. | |
| From: Peter Simons (Parts [1987], 6.3) | |
| A reaction: This sounds blatantly false. If you know something is a 'snake', that doesn't tell you how hot it must get before the snakes die. Obviously if you know all about snakes (from studying individual snakes!), then you know a lot about the next snake. |
| 12886 | A whole requires some unique relation which binds together all of the parts [Simons] |
| Full Idea: A whole must at least approximate to this condition: every member of some division of the object stands in a certain relation to every other member, and no member bears this relation to anything other than members of the division. | |
| From: Peter Simons (Parts [1987], 9.2) | |
| A reaction: Simons proceeds to formalise this, and I suspect that he goes for this definition because (unlike looser ones) it can be formalised. See Simons's Idea 12865. We'll need to know whether these are internal or external relations. |
| 12835 | Does Tibbles remain the same cat when it loses its tail? [Simons] |
| Full Idea: The cat is 'Tibbles' with a tail; 'Tib' is Tibbles after the loss of the tail. 1) Tibbles isn't Tib at t; 2) Tibbles is Tib at t'; 3) Tibbles at t is Tibbles at t'; 4) Tib at t is Tib at t'; so 5) Tibbles at t is Tib at t (contradicting 1). What's wrong? | |
| From: Peter Simons (Parts [1987], 3.3) | |
| A reaction: [The example is in Wiggins 1979, from Geach, from William of Sherwood] Simons catalogues nine assumptions which are being made to produce the contradiction. 1) rests on Leibniz's law. Simons says two objects are occupying Tibbles. |
| 12857 | Tibbles isn't Tib-plus-tail, because Tibbles can survive its loss, but the sum can't [Simons] |
| Full Idea: There mere fact that Tibbles can survive the mutilation of losing a tail, whereas the sum of Tib and the tail cannot, is enough to distinguish them, even if no such mutilation ever occurs. | |
| From: Peter Simons (Parts [1987], 6.1) | |
| A reaction: See Idea 12835 for details of the Tibbles example. Either we go for essentialism here, or the whole notion of identity collapses. But the essential features of a person are not just those whose loss would kill them. |
| 12820 | Without extensional mereology two objects can occupy the same position [Simons] |
| Full Idea: If we reject extensionality in mereology, it has as a consequence that more than one object may have exactly the same parts at the same time, and hence occupy the same position. | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: Simons defends this claim. I'm unconvinced that we must choose between the two views. The same parts should ensure the same physical essence, which seems to guarantee the same identity. Not any old parts generate an essence. |
| 12866 | Composition is asymmetric and transitive [Simons] |
| Full Idea: Composition is asymmetric and transitive: if a is made up of b, and b of c, then a is made up of c; and if a is made of b, then b is not made up of a. We cannot say the snow is made up of the snowball. | |
| From: Peter Simons (Parts [1987], 6.5) | |
| A reaction: ...And snowballs composed of snow can then compose a snowman (transitivity). |
| 12867 | A hand constitutes a fist (when clenched), but a fist is not composed of an augmented hand [Simons] |
| Full Idea: Composition entails constitution, but does the converse hold? A hand constitutes a fist in virtue of being clenched, but it is not obvious that it composes a fist, and certainly a fist is not composed of a hand plus some additional part. | |
| From: Peter Simons (Parts [1987], 6.5) | |
| A reaction: There are subtleties of ordinary usage in 'compose' and 'constitute' which are worth teasing apart, but that isn't the last word on such relationships. 'Compose' seems to point towards matter, while 'constitute' seems to point towards form. |
| 12864 | We say 'b is part of a', 'b is a part of a', 'b are a part of a', or 'b are parts of a'. [Simons] |
| Full Idea: There are four cases of possible forms of expression when a is made up of b: we say 'b is part of a', or 'b is a part of a', or 'b are a part of a', or 'b are parts of a'. | |
| From: Peter Simons (Parts [1987], 6.4) | |
| A reaction: Personally I don't want to make much of these observations of normal English usage, but they are still interesting, and Simons offers a nice discussion of them. |
| 12814 | Classical mereology says there are 'sums', for whose existence there is no other evidence [Simons] |
| Full Idea: Either out of conviction or for reasons of algebraic neatness, classical extensional mereology asserts the existence of certain individuals, mereological sums, for whose existence in general we have no evidence outside the theory itself. | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: Observing that we have no evidence for sums 'outside the theory' is nice. It is a nice ontological test, with interesting implications for Quinean ontological commitment. |
| 12817 | 'Mereological extensionality' says objects with the same parts are identical [Simons] |
| Full Idea: Classical extensional mereology won't extend well to temporal and modal facts, because of 'mereological extensionality', which is the thesis that objects with the same parts are identical (by analogy with the extensionality of sets). | |
| From: Peter Simons (Parts [1987], Intro) | |
| A reaction: Simons challenges this view, claiming, for example, that the Ship of Theseus is two objects rather than one. I suppose 'my building bricks' might be 'your sculpture', but this is very ontologically extravagant. This is a mereological Leibniz's Law. |
| 12833 | If there are c atoms, this gives 2^c - 1 individuals, so there can't be just 2 or 12 individuals [Simons] |
| Full Idea: In classical mereology, if there are c atoms, where c is any cardinal number, there are 2^c - 1 individuals, so the cardinality of models is restricted. There are no models with cardinality 2, 12 or aleph-0, for example. | |
| From: Peter Simons (Parts [1987], 1.2) | |
| A reaction: The news that there is no possible world containing just 2 or just 12 individuals ought to worry fans of extensional mereology. A nice challenge for God - create a world containing just 12 individuals. |
| 12849 | Sums are more plausible for pluralities and masses than they are for individuals [Simons] |
| Full Idea: We are on stronger grounds in asserting the general existence of sums when considering pluralities and masses than when considering individuals. | |
| From: Peter Simons (Parts [1987], 5.2) | |
| A reaction: I was thinking that the modern emphasis on referring to plurals was precisely to resist the idea that we must 'sum' them into one thing. If so, we wouldn't want to then sum several plurals. If a mass isn't a sum, how can we sum some masses? |
| 12877 | Sums of things in different categories are found within philosophy. [Simons] |
| Full Idea: Cross-categorial sums are not unknown in philosophy. A body and the events which befall it are intimately connected, and the mysterious four-dimensional blocks might be mereological sums of the body and its life. | |
| From: Peter Simons (Parts [1987], 8.1) | |
| A reaction: Simons here ventures into the territory of abstracta, which he said he wouldn't touch. Presumably his first example has 'a biography' as its whole, which is not just a philosophical notion. Why will some categories sum, and others won't? |
| 12888 | The wholeness of a melody seems conventional, but of an explosion it seems natural [Simons] |
| Full Idea: The example of a melody shows that what counts as a temporal individual is partly a matter of human stipulation. But with a natural event like an explosion there is little or no room for decision about what is a part, and whether it is a single event. | |
| From: Peter Simons (Parts [1987], 9.6) | |
| A reaction: You could have a go at giving a natural account of the wholeness of a melody, in terms of the little aesthetic explosion that occurs in the brain of a listener. |
| 12871 | Objects have their essential properties because of the kind of objects they are [Simons] |
| Full Idea: An object has the essential properties it has in virtue of being the kind of object it is. | |
| From: Peter Simons (Parts [1987], 7.1) | |
| A reaction: He attributes this to Husserl and Wiggins. I just don't get it. What makes something the 'kind of object it is'? They've got it the wrong way round. Does God announce that this thing is a tiger, and is then pleasantly surprised to discover its stripes? |
| 12870 | We must distinguish the de dicto 'must' of propositions from the de re 'must' of essence [Simons] |
| Full Idea: We must distinguish the 'must' of necessity as applied to a proposition or state of affairs (de dicto) from the 'must' of essence, concerning the way in which an object has an attribute (de re). | |
| From: Peter Simons (Parts [1987], 7.1) | |
| A reaction: A helfpful distinction, but a possible confusion of necessity and essentiality (Simons knows this). Modern logicians seem to run them together, because they only care about identity. I don't, because I care about explanations. |
| 12873 | Original parts are the best candidates for being essential to artefacts [Simons] |
| Full Idea: Original parts are the best candidates for being essential to artefacts. It is hard to conceive how an object could have as essential a part which was attached at some time after the object had come into being. | |
| From: Peter Simons (Parts [1987], 7.4) | |
| A reaction: Without its big new memory upgrade my computer would be hopelessly out of date. Simons is awesome in some ways, but seems rather confused when it comes to discussing essence. I think Wiggins may have been a bad influence on him. |
| 12874 | An essential part of an essential part is an essential part of the whole [Simons] |
| Full Idea: An essential part of an essential part is an essential part of the whole. | |
| From: Peter Simons (Parts [1987], 7.4) | |
| A reaction: Sounds beyond dispute, but worth pondering. It seems to be only type-parts, not token-parts, which are essential. Simons is thinking of identity rather than function, but he rejects Chisholm's idea that all parts are essential. So which ones are? |
| 12837 | Four dimensional-objects are stranger than most people think [Simons] |
| Full Idea: The strangeness of four-dimensional objects is almost always underestimated in the literature. | |
| From: Peter Simons (Parts [1987], 3.4) | |
| A reaction: See Idea 12836, where he has criticised process ontologists for smuggling in stages and process as being OF conventional objects. |
| 12856 | Intermittent objects would be respectable if they occurred in nature, as well as in artefacts [Simons] |
| Full Idea: If we could show that intermittence could occur not only among artefacts and higher-order objects, but also among natural things, then we should have given it a secure place on the ontological map. | |
| From: Peter Simons (Parts [1987], 5.7) | |
| A reaction: Interesting ontological test. Having identified fairly clear intermittent artefacts (Idea 12851), if we then fail to find any examples in nature, must we revisit the artefacts and say they are not intermittents? He suggests freezing an organ in surgery. |
| 12885 | Objects like chess games, with gaps in them, are thereby less unified [Simons] |
| Full Idea: Temporal objects which are scattered in time - i.e. have temporal gaps in them, like interrupted discussions or chess games - are less unified than those without gaps. | |
| From: Peter Simons (Parts [1987], 9.2) | |
| A reaction: Is he really saying that a discussion or a chess game is less unified if there is even the slightest pause in it? Otherwise, how long must the pause be before it disturbs the unity? Do people play internet chess, as they used to play correspondence chess? |
| 12854 | An entrepreneur and a museum curator would each be happy with their ship at the end [Simons] |
| Full Idea: At the end of the Ship of Theseus story both an entrepreneur and a museum curator can be content, each having his ship all to himself, ..because each was all along claiming a different object from the other. | |
| From: Peter Simons (Parts [1987], 5.5) | |
| A reaction: Simons has the entrepreneur caring about function (for cruises), and the curator caring about matter (as a relic of Theseus). It is bold of Simons to say on that basis that it starts as two objects, one 'matter-constant', the other 'form-constant'. |
| 12855 | The 'best candidate' theories mistakenly assume there is one answer to 'Which is the real ship?' [Simons] |
| Full Idea: The 'best candidate' theories get into difficulty because it is assumed that there is a single uniquely correct answer to the question 'Which is the real ship?' | |
| From: Peter Simons (Parts [1987], 5.5) | |
| A reaction: My own example supports Simons. If Theseus discards the old planks as rubbish, then his smart new ship is the original. But if he steals his own ship (to evade insurance regulations) by substituting a plank at a time, the removed planks are the original. |
| 12872 | The zygote is an essential initial part, for a sexually reproduced organism [Simons] |
| Full Idea: It is essential to an organism arising from sexual reproduction that it has its zygote as initial improper part. | |
| From: Peter Simons (Parts [1987], 7.3) | |
| A reaction: It can't be necessary that an organism which appears to be sexually reproduced actually is so (if you don't believe that, read more science fiction). It may well just be analytic that sexual reproduction involves a zygote. Nothing to do with essence. |
| 12889 | The limits of change for an individual depend on the kind of individual [Simons] |
| Full Idea: What determines the limits of admissible change and secures the identity of a continuant is a matter of the kind of object in question. | |
| From: Peter Simons (Parts [1987], 9.6) | |
| A reaction: This gives some motivation for the sortal view of essence, which I find hard to take. However, if my statue were pulverised it would make good compost. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 10432 | A new usage of a name could arise from a mistaken baptism of nothing [Sainsbury] |
| Full Idea: A baptism which, perhaps through some radical mistake, is the baptism of nothing, is as good a propagator of a new use as a baptism of an object. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.3) | |
| A reaction: An obvious example might be the Loch Ness Monster. There is something intuitively wrong about saying that physical objects are actually part of linguistic meaning or reference. I am not a meaning! |
| 10434 | Even a quantifier like 'someone' can be used referentially [Sainsbury] |
| Full Idea: A large range of expressions can be used with referential intentions, including quantifier phrases (as in 'someone has once again failed to close the door properly'). | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.5) | |
| A reaction: This is the pragmatic aspect of reference, where it can be achieved by all sorts of means. But are quantifiers inherently referential in their semantic function? Some of each, it seems. |
| 12843 | With activities if you are doing it you've done it, with performances you must finish to have done it [Simons] |
| Full Idea: Action theorists distinguish between activity verbs such as 'weep' and 'talk' (where continuous entails perfect - John is weeping so John has now wept), and performance verbs like 'wash', where John is washing doesn't yet mean John has washed. | |
| From: Peter Simons (Parts [1987], 4.2) | |
| A reaction: How to distinguish them, bar examples? In 'has wept' and 'has washed', I'm thinking that it is the 'has' which is ambiguous, rather than the more contentful word. One is 'has participated' and the other is 'has completed'. I've participated in washing! |
| 12875 | One false note doesn't make it a performance of a different work [Simons] |
| Full Idea: A performance of a certain work with a false note is still a performance of that work, albeit a slightly imperfect one, and not (as Goodman has argued) a performance of a different work. | |
| From: Peter Simons (Parts [1987], 7.6) | |
| A reaction: This is clearly right, but invites the question of how many wrong notes are permissable. One loud very wrong note could ruin a very long performance (but of that work, presumably). This is about classical music, but think about jazz. |
| 10431 | Things are thought to have a function, even when they can't perform them [Sainsbury] |
| Full Idea: On one common use of the notion of a function, something can possess a function which it does not, or even cannot, perform. A malformed heart is to pump blood, even if such a heart cannot in fact pump blood. | |
| From: Mark Sainsbury (The Essence of Reference [2006], 18.2) | |
| A reaction: One might say that the heart in a dead body had the function of pumping blood, but does it still have that function? Do I have the function of breaking the world 100 metres record, even though I can't quite manage it? Not that simple. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |